C₆₁×#12804 #29516237

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#12804

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

432 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 97 points:

(2,4)(5,6,10,15,11,16,8,20)(7,12,17,27,19,28,21,35)(9,14,22,30,24,31,23,13)(18,26,33,25,34,36,32,29)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,3,4)(5,7)(6,12)(8,21)(9,18)(10,17)(11,19)(13,29)(14,26)(15,27)(16,28)(20,35)(22,33)(23,32)(24,34)(25,30)(31,36);(5,8,11,10)(6,13,16,30)(7,17,19,21)(9,23,24,22)(12,25,28,29)(14,15,31,20)(18,33,34,32)(26,35,36,27)

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5792);